= Affine-target adjunction for varieties
{title2=$\operatorname{Hom}(X,Y)\cong\operatorname{Hom}_{k\text{-alg}}(k[Y],\Gamma(X,\mathcal O_X))$}
For a classical <algebraic variety> $X$ and <affine variety> $Y$, a homomorphism on the indicated rings gives a morphism by evaluating the images of affine coordinate generators at each point of $X$. The relations defining $Y$ remain zero. Local fractions pull back to local regular fractions, proving regularity; the <coordinate ring> generators also prove uniqueness.
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